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A modern maximum-likelihood theory for high-dimensional logistic regression.

Pragya Sur1, Emmanuel J Candès1,2

  • 1Department of Statistics, Stanford University, Stanford, CA 94305; candes@stanford.edu pragya@stanford.edu.

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|July 3, 2019
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Summary
This summary is machine-generated.

Statistical inferences from logistic models with many variables are often unreliable. This study reveals that maximum likelihood estimates are biased and have greater variability than assumed, impacting predictions and statistical tests.

Keywords:
high-dimensional inferencelikelihood-ratio testlogistic regressionmaximum-likelihood estimate

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Area of Science:

  • Statistics
  • Data Science
  • Machine Learning

Background:

  • Traditional statistical inference for logistic models assumes large sample sizes relative to the number of variables.
  • Commonly accepted rules of thumb (5-10 observations per parameter) for reliable inference are often insufficient.
  • Existing methods for quantifying estimate variability and performing statistical inference may be inaccurate in high-dimensional settings.

Purpose of the Study:

  • To investigate the validity of classical statistical inference for logistic models when the number of variables (p) grows with sample size (n).
  • To demonstrate that maximum likelihood estimates (MLE) and likelihood-ratio tests (LRT) can be unreliable under these conditions.
  • To develop a new theoretical framework for accurate inference in high-dimensional logistic regression.

Main Methods:

  • Analysis of logistic models with independent features where n and p increase in a fixed ratio.
  • Derivation of explicit expressions for the asymptotic bias and variance of the MLE.
  • Determination of the asymptotic distribution of the LRT.
  • Empirical validation of theoretical results in finite samples.

Main Results:

  • The maximum likelihood estimate (MLE) is asymptotically biased.
  • The variability of the MLE is significantly greater than classically estimated.
  • The likelihood-ratio test (LRT) is not asymptotically distributed as a chi-squared (χ²) distribution.
  • Bias in MLE leads to inaccurate probability predictions.

Conclusions:

  • Classical inference methods for logistic regression are often unreliable when the number of variables is large relative to the sample size.
  • A new theory provides accurate expressions for bias, variance, and LRT distribution, applicable in finite samples.
  • Inference accuracy depends on a signal strength measure, enabling concrete proposals for reliable finite-sample inference.